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Vectors Proof Questions worksheet

8 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

8 questions 39 marks Sheet 1
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  1. 5 marks

    \(OAB\) is a triangle.

    Triangle OAB, not to scale, with the side from O to A drawn as an arrow labelled 2a and the side from O to B drawn as an arrow labelled 6b, and a line drawn from X on OA to Y on OB2a6bOABXYDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 2\mathbf{a}\) and \(\overrightarrow{OB} = 6\mathbf{b}\).

    \(X\) is the point on \(OA\) such that \(OX : XA = 1 : 1\), and \(Y\) is the point on \(OB\) such that \(OY : YB = 1 : 1\).

    (a)

    Write \(\overrightarrow{XY}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Show that \(XY\) is parallel to \(AB\), and write down the ratio \(XY : AB\).

  2. 7 marks

    \(OABC\) is a parallelogram. \(X\) is the midpoint of the diagonal \(AC\).

    Parallelogram OABC, not to scale, with the side from O to A drawn as an arrow labelled 6a, the side from O to C drawn as an arrow labelled 4b, the diagonal AC drawn, and X marked at its midpoint6a4bOABCXDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 6\mathbf{a}\) and \(\overrightarrow{OC} = 4\mathbf{b}\).

    (a)

    Express \(\overrightarrow{AC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Express \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (c)

    Prove that \(O\), \(X\) and \(B\) lie on the same straight line.

  3. 1 mark

    The points \(A\), \(B\) and \(C\) have position vectors, relative to an origin \(O\),

    \(\overrightarrow{OA} = 4\mathbf{a} + 2\mathbf{b}\)

    \(\overrightarrow{OB} = 6\mathbf{a} + 3\mathbf{b}\)

    \(\overrightarrow{OC} = 12\mathbf{a} + 6\mathbf{b}\)

    where \(\mathbf{a}\) and \(\mathbf{b}\) are not parallel.

    Prove that \(A\), \(B\) and \(C\) lie on the same straight line.

  4. 6 marks

    \(OABC\) is a trapezium. \(M\) is the midpoint of \(BC\).

    Trapezium OABC, not to scale, with O to A drawn as an arrow labelled 4a, A to B as an arrow labelled 4b, O to C as an arrow labelled 12b, and M marked halfway along BC4a4b12bOABCMDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 4\mathbf{a}\), \(\overrightarrow{AB} = 4\mathbf{b}\) and \(\overrightarrow{OC} = 12\mathbf{b}\).

    (a)

    Find \(\overrightarrow{BC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Find \(\overrightarrow{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

  5. 7 marks

    \(OABC\) is a parallelogram. \(P\) is the midpoint of \(AB\) and \(Q\) is the midpoint of \(BC\).

    Parallelogram OABC, not to scale, with O to A drawn as an arrow labelled 6a and O to C as an arrow labelled 4b, the diagonal AC drawn, and the midpoints P of AB and Q of BC joined6a4bOABCPQDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 6\mathbf{a}\) and \(\overrightarrow{OC} = 4\mathbf{b}\).

    (a)

    Express \(\overrightarrow{AC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Express \(\overrightarrow{PQ}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (c)

    Show that \(PQ\) is parallel to \(AC\), and write down the ratio \(PQ : AC\).

  6. 5 marks

    \(OAB\) is a triangle.

    Triangle OAB, not to scale, with the side from O to A drawn as an arrow labelled 6a and the side from O to B drawn as an arrow labelled 9b, and a line drawn from X on OA to Y on OB6a9bOABXYDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 6\mathbf{a}\) and \(\overrightarrow{OB} = 9\mathbf{b}\).

    \(X\) is the point on \(OA\) such that \(OX : XA = 1 : 2\), and \(Y\) is the point on \(OB\) such that \(OY : YB = 1 : 2\).

    (a)

    Find \(\overrightarrow{XY}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Show that \(XY\) is parallel to \(AB\), and write down the ratio \(XY : AB\).

  7. 7 marks

    \(OABC\) is a parallelogram. \(X\) is the midpoint of the diagonal \(AC\).

    Parallelogram OABC, not to scale, with the side from O to A drawn as an arrow labelled 10a, the side from O to C drawn as an arrow labelled 8b, the diagonal AC drawn, and X marked at its midpoint10a8bOABCXDiagram NOT accurately drawn

    \(\overrightarrow{OA} = 10\mathbf{a}\) and \(\overrightarrow{OC} = 8\mathbf{b}\).

    (a)

    Find \(\overrightarrow{AC}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Find \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (c)

    Prove that \(O\), \(X\) and \(B\) lie on the same straight line.

  8. 1 mark

    Relative to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors

    \(\overrightarrow{OA} = 2\mathbf{a} + 3\mathbf{b}\)

    \(\overrightarrow{OB} = 11\mathbf{a} + 6\mathbf{b}\)

    \(\overrightarrow{OC} = 14\mathbf{a} + 7\mathbf{b}\)

    where \(\mathbf{a}\) and \(\mathbf{b}\) are not parallel.

    Prove that \(A\), \(B\) and \(C\) lie on the same straight line.

Answers

  1. (a) \(3\mathbf{b} - \mathbf{a}\)
    (b) \(\overrightarrow{AB} = 6\mathbf{b} - 2\mathbf{a} = 2\left(3\mathbf{b} - \mathbf{a}\right) = 2\,\overrightarrow{XY}\), so \(XY\) is parallel to \(AB\), and \(XY : AB = 1 : 2\)
  2. (a) \(4\mathbf{b} - 6\mathbf{a}\)
    (b) \(3\mathbf{a} + 2\mathbf{b}\)
    (c) \(\overrightarrow{OB} = 6\mathbf{a} + 4\mathbf{b} = 2\left(3\mathbf{a} + 2\mathbf{b}\right) = 2\,\overrightarrow{OX}\), so \(OB\) is parallel to \(OX\) and they share the point \(O\); therefore \(O\), \(X\) and \(B\) are on one straight line
  3. \(\overrightarrow{AB} = 2\mathbf{a} + \mathbf{b}\) and \(\overrightarrow{BC} = 6\mathbf{a} + 3\mathbf{b} = 3\,\overrightarrow{AB}\), so \(AB\) and \(BC\) are parallel and share the point \(B\); therefore \(A\), \(B\) and \(C\) are on one straight line
  4. (a) \(8\mathbf{b} - 4\mathbf{a}\)
    (b) \(2\mathbf{a} + 8\mathbf{b}\)
  5. (a) \(4\mathbf{b} - 6\mathbf{a}\)
    (b) \(2\mathbf{b} - 3\mathbf{a}\)
    (c) \(\overrightarrow{AC} = 4\mathbf{b} - 6\mathbf{a} = 2\left(2\mathbf{b} - 3\mathbf{a}\right) = 2\,\overrightarrow{PQ}\), so \(PQ\) is parallel to \(AC\), and \(PQ : AC = 1 : 2\)
  6. (a) \(3\mathbf{b} - 2\mathbf{a}\)
    (b) \(\overrightarrow{AB} = 9\mathbf{b} - 6\mathbf{a} = 3\left(3\mathbf{b} - 2\mathbf{a}\right) = 3\,\overrightarrow{XY}\), so \(XY\) is parallel to \(AB\), and \(XY : AB = 1 : 3\)
  7. (a) \(8\mathbf{b} - 10\mathbf{a}\)
    (b) \(5\mathbf{a} + 4\mathbf{b}\)
    (c) \(\overrightarrow{OB} = 10\mathbf{a} + 8\mathbf{b} = 2\left(5\mathbf{a} + 4\mathbf{b}\right) = 2\,\overrightarrow{OX}\), so \(OB\) is parallel to \(OX\) and they share the point \(O\); therefore \(O\), \(X\) and \(B\) are on one straight line
  8. \(\overrightarrow{AB} = 9\mathbf{a} + 3\mathbf{b}\) and \(\overrightarrow{BC} = 3\mathbf{a} + \mathbf{b} = \frac{1}{3}\,\overrightarrow{AB}\), so \(AB\) and \(BC\) are parallel and share the point \(B\); therefore \(A\), \(B\) and \(C\) are on one straight line

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